“Pre-Lie algebra”的意思、由来-开放百科全书

词条

Pre-Lie algebra

释义

Definition
Examples
Vector fields on an affine space Rooted trees
References

In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space.

The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras.

Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.

Definition

A pre-Lie algebra is a vector space with a bilinear map , satisfying the relation

This identity can be seen as the invariance of the associator under the exchange of the two variables and .

Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically. Although weaker than associativity, the defining relation of a pre-Lie algebra still implies that the commutator is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the terms in the defining relation for pre-Lie algebras, above.

Examples

Vector fields on an affine space

Let be an open neighborhood of , parameterised by variables . Given vector fields , we define .

The difference between and , is

which is symmetric in and . Thus defines a pre-Lie algebra structure.

Given a manifold and homeomorphisms from to overlapping open neighborhoods of , they each define a pre-Lie algebra structure on vector fields defined on the overlap. Whilst need not agree with , their commutators do agree: , the Lie bracket of and .

Rooted trees

Let be the free vector space spanned by all rooted trees.

One can introduce a bilinear product on as follows. Let and be two rooted trees.

where is the rooted tree obtained by adding to the disjoint union of and an edge going from the vertex of to the root vertex of .

Then is a free pre-Lie algebra on one generator.

References

{{citation

| last1 = Chapoton | first1 = F.
| last2 = Livernet | first2 = M.
| year = 2001
| mr = 1827084
| journal = International Mathematics Research Notices
| title = Pre-Lie algebras and the rooted trees operad
| doi = 10.1155/S1073792801000198
| pages = 395–408
| volume = 8
| issue = 8}}.

{{citation

| last = Szczesny | first = M.
| year = 2010
| mr =
| journal =
| title = Pre-Lie algebras and incidence categories of colored rooted trees
| volume =1007
| bibcode = 2010arXiv1007.4784S
| pages = 4784
| arxiv = 1007.4784 }}.

3 : Lie groups|Lie algebras|Non-associative algebra

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。